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6. Three-body motions in the equilateral configuration

Three bodies coupled by mutual gravitational forces can synchronously execute surprisingly simple (Keplerian) motions. This section shows several examples of possible regular three-body motions in the equilateral configuration.

Examples:

1. Circular motion of three bodies in the case of equal masses. Three bodies of equal masses coupled by mutual gravitation can move along a common circle that circumscribes the equilateral triangle formed by the bodies. In this case the triangle rotates uniformly about its centre as a rigid body.

2. Instability of the circular motion of three bodies in the case of equal masses. The circular motion of the previous example is unstable with respect to arbitrarily small perturbations of positions or velocities that destroy the symmetry. In this example the initial velocity of one of the bodies slightly differ from the value that is necessary for the circular motion, and this tiny violation of the symmetry causes a distortion that increases progressively. The motion remains reversible until the bodies collide. (Click also here to see the applet.)

3. Elliptical motion of three bodies in the case of equal masses. If the initial velocities of the bodies, being equal in magnitudes (and perpendicular to the radius vectors of bodies), differ from the value that provides the circular motion, the bodies synchronously trace congruent ellipses whose axes make angles of 120 degrees with one another. In this case the equilateral triangle rotates nonuniformly about its centre, and the length of its sides vary periodically. (Click also here to see the applet.)

4. Instability of elliptical motions of three bodies in the case of equal masses. To illustrate the instability of motion of the previous example, here the initial velocity of one of the bodies is slightly changed. This tiny violation of the symmetry causes a distortion that increases progressively. The motion remains reversible until the bodies collide. (Click also here to see the applet.)

5. Circular motion of three bodies in the case of different masses. Three bodies of different masses coupled by mutual gravitational forces can move synchronously along concentric circles. In this case the triangle rotates uniformly as a rigid body about the centre of mass of the system. (Click also here to see the applet.)

6. Instability of the circular motion of three bodies in the case of different masses. A tiny variation of the initial velocity of one of the bodies from the value of the previous example violates slightly the symmetry of the system causing a distortion that increases progressively. (Click also here to see the applet.)

7. Elliptical motion of three bodies in the case of different masses. Three bodies of different masses coupled by mutual gravitation can move synchronously along conic sections in the equilateral configuration. In this case the equilateral triangle joining the bodies rotates nonuniformly, and the length of its sides vary periodically during the motion. (Click also here to see the applet.)

8. Instability of elliptical motions of three bodies in the case of different masses. Again a tiny variation of the initial velocity of one of the bodies from the value of the previous example allows to illustrate the instability of this motion. The motion remains reversible until the bodies collide - if we reverse velocities of all the bodies, the backward motion soon restores the initial equilateral configuration. (Click also here to see the applet.)

9. Satellite at the triangular libration point of two massive bodies that execute circular motions. It is possible to launch such a stationary satellite to the triangular libration points in the Earth - Moon system. The program simulates an imaginary system with m/M = 0.12. (Click also here to see the applet.)

Explanation

To better understand why such a simple motion can occur in a three-body system whose behaviour generally is very complicated and cannot be described by an analytical solution to the equations of motion, we should start with the simplest possible example. If three bodies of equal masses are located at the vertices of an equilateral triangle, the net force exerted on each of the bodies by the gravitation of the other two bodies is directed towards the centre of this triangle. The magnitude of this force is inversely proportional to the square of the distance of the body from this centre. That is, each body can behave as if it were subjected to the gravitational force produced by a single immovable point source (with some effective gravitational mass) located at the centre of the triangle rather than to the gravitational forces exerted by the other two moving bodies. The same is true for all the three bodies, provided the equilateral configuration is preserved during their motion. Therefore the bodies can synchronously execute Keplerian motions along similar (homothetic) conic sections. In particular, they can move along a common circle circumscribing the equilateral triangle formed by the bodies. In this case the triangle rotates uniformly about its centre. Click here (and once more here to see the applet) to observe an example of such a motion.

The circular motion in the equilateral configuration occurs if all the three bodies (of equal masses) have initial velocities directed perpendicularly to their radius vectors. These velocities must have equal and quite definite magnitudes. However, if equal magnitudes of the initial velocities differ from this value, and/or the velocities are not perpendicular to the radius vectors (but directed for all the bodies at the same angle with the radius vector), the bodies will synchronously trace congruent ellipses (generally -- conic sections) whose axes make angles of 120 degrees with one another. Click here (and once more here to see the applet) to observe an example of such a motion.

When all three bodies have equal masses, as in the preceding examples, we can easily reconcile with our intuition the possibility of their motion in the equilateral configuration appealing to the symmetry of the system. However, the equilateral configuration can be preserved during the motion even when the masses of the bodies are different. This is possible because in the equilateral configuration the total gravitational force exerted on each of the bodies by the other two bodies is directed towards the centre of mass of the system and is inversely proportional to the square of the distance from the centre of mass. In other words, each of the three bodies coupled by mutual gravitational forces can be considered as moving in an effective stationary central inverse square gravitational field with the source at the centre of mass of the system, although this field is produced by the other two moving bodies. This stationary source of the central field is characterized by some effective gravitational mass. Hence the equilateral configuration of the system can be preserved during the motion, in which the bodies trace synchronously homothetic Keplerian ellipses whose common focus is located at the centre of mass of the system. Linear dimensions of these ellipses are inversely proportional to the masses of the bodies. In this case the equilateral triangle joining the bodies rotates nonuniformly, and the length of its sides vary periodically during the motion. A rigorous proof can be found in the paper by Eugene Butikov Regular Keplerian Motions in Classical Many-Body Systems, European Journal of Physics, v. 21, pp. 465 482, 2000. In particular, the bodies can move in the equilateral configuration along concentric circles. Click here (and once more here to see the applet) to observe an example of such a motion.

Click here (and once more here to see the applet) in order to observe an example of synchronous elliptical motions of three different bodies in the equilateral configuration. Linear dimensions of these ellipses are inversely proportional to the masses of bodies.

Next section deals with possible regular planar motions of four bodies of equal masses in the equilateral configuration.

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